Diffeomorphisms of the Worldsheet

String Theory without the Baggage, episode 19

I’m as excited to share this photo of some cliffs near the Colorado River as I am to finally be dealing with an actual string action today!

When studying the relativistic point particle, we found that the naive global symmetry of time-translation invariance was merely a residual of a broader, reparametrization symmetry of the world line.

Thinking of the worldline as a one-dimensional manifold, we found that the theory was invariant under diffeomorphisms of that worldline¹. This fact brought forth all kinds of complications in the quantum theory, as we were suddenly faced with a gauge symmetry.

The worldsheet of the string makes things more complex. Diffeomorphisms of the worldsheet now involves two coordinates.

Of course, this generalizes higher-dimensional objects, and that is a big problem. General Relativity is by definition a gauge theory of coordinate diffeomorphisms, and the whole point of String Theory is an attempt to quantize it.

Fortunately, dimension matters. The gravitational field is only dynamical for spacetimes with dimension four or above². For the two-dimensional case of the worldsheet, we shall find additional symmetry that makes quantization tractable.

The proper treatment of these local symmetries in passing to the quantum theory is what makes String Theory a fascinating mathematical apparatus in its own right.

In this episode we will present the main descriptive logic of the classical theory, and will treat the precise details in a few follow up episodes.

The Polyakov Action (n=2)

The action to embed a worldsheet Σ into (flat) spacetime is

\(S = \kappa \int d^{2}\sigma \, \sqrt{-\gamma} \left( \frac{1}{2}\gamma^{ab}g_{\mu\nu} \partial_{a}X^{\mu}\partial_{b}X^{\nu} + \Lambda + R^{(2)}(\gamma)\right).\)

Here 𝛾 is our auxiliary metric on Σ. Generically, the kinetic sector for 𝛾 is encoded in the Ricci scalar. Like the one-dimensional case, R^(2) won't play a direct role in the worldsheet action. In two-dimensions, R^(2) is a total-derivative, making it relevant only for boundary concerns³. For us this means string interactions. But we need to study the free string first.

Without a kinetic term, the metric 𝛾 is akin to a Lagrange multiplier. It is present to enforce constraints. It is interesting to note in two-dimensions, these constraints include the fact the worldsheet cosmological constant, Λ vanishes⁴.

Therefore our action reduces to

\(S = \frac{\kappa}{2}\int d^{2}\sigma \, \sqrt{-\gamma} \, \gamma^{ab}g_{\mu\nu} \partial_{a}X^{\mu}\partial_{b}X^{\nu} .\)

Local Symmetries of the Worldsheet Action

As with the worldine, we can represent an infinitesimal diffeomorphism as coordinate change:

\(\sigma^{a} \rightsquigarrow \sigma^{a} + \xi^{a}.\)

It is local in the sense that

\(\xi^{a} = \xi^{a}(\sigma).\)

In line with this, the metric varies,

\(\gamma_{ab}\rightsquigarrow \gamma_{ab} + \nabla_{a}\,\xi_{b} + \nabla_{b}\,\xi_{a},\)

where ∇ is the covariant derivative associated to 𝛾. That is, the Levi-Civita connection on the worldsheet.

One can show⁵ that this implies the infinitesimal changes in X other metric related objects are given as

\(\begin{eqnarray*} \delta X^{\mu} &=& \xi^{a}\partial_{a}X^{\mu},\\ \delta \gamma^{ab} &=& \xi^{c}\partial_{c}\gamma^{ab} - \partial^{a}\xi^{b} - \partial^{b}\xi^{a},\\ \delta\sqrt{-\gamma} &=& \partial_{a}\left(\xi^{a}\sqrt{-\gamma}\right). \end{eqnarray*}\)

In two-dimensions we have an additional symmetry⁶ involving local rescaling of the metric:

\(\gamma_{ab} \rightsquigarrow \Omega(\sigma) \,\gamma_{ab}\)

To see this⁷, let’s write this scaling in terms of a worldsheet function φ:

\(\Omega(\sigma) = e^{\phi(\sigma)}.\)

The determinant of the metric then scale as

\(\sqrt{-\gamma} \rightsquigarrow \sqrt{-\gamma\, e^{2\phi}} = e^{\phi}\sqrt{-\gamma},\)

and the inverse metric also scales inversely

\(\gamma^{ab}\rightsquigarrow e^{-\phi}\gamma^{ab},\)

which demonstrates this additional scale or Weyl invariance⁸.

The upshot of these three local symmetries is that we can pick coordinates that lets us select each of the three independent values of the auxiliary metric 𝛾. The obvious choice is the flat metric, which further simplifies our action

\(S = \frac{\kappa}{2}\int d^{2}\sigma \, \partial_{a}X^{\mu}\partial^{a}X_{\mu} .\)

In the relativistic point particle case, setting 𝜆 = 1 was something of a classical nicety⁹. For the relativistic string, flattening the metric 𝛾 is crucial to make the theory tractable, even classically.

We will of course pay the price for this gauge choice with the introduction of BRST ghosts in the quantum theory.

Nevertheless, varying the action with respect to X now gives the field equations.

The Constraint Equations

These field equations aren’t the end of the story. They’re still constrained by geometry. These are somewhat similar to the mass-shell condition of the relativistic point particle¹⁰. To determine the constraints, we must vary the action with respect to 𝛾. Of course, consistency requires that we must do this before we impose our gauge choice.

It is traditional to give the variation of S with respect to 𝛾 as a worldsheet stress-energy tensor. This turns out to be

\(T^{ab} \propto \frac{\delta S}{\delta \gamma_{ab}} \propto \partial^{a}X^{\mu}\partial^{b}X_{\mu} - \frac{1}{2}\gamma^{ab}\partial^{m}X^{\mu}\partial_{m}X_{\mu}\)

The constraint equations are just

\(T^{ab} = 0.\)

This gives three distinct equations, one for each coordinate. Carefully tracking the metric’s minus signs reveal that the diagonal constraints are the same:

\(T^{00} = T^{11}= 0 \Rightarrow \partial_{0}X^{\mu}\partial_{0}X_{\mu} + \partial_{1}X^{\mu}\partial_{1}X_{\mu} = 0.\)

Given that T is symmetric, the diagonal terms are of course equivalent:

\(T^{01} = 0 \Rightarrow \partial_{0}X^{\mu}\partial_{1}X_{\mu} = 0.\)

The Equations of Motion

Varying our simplified action with respect to the embedding X gives the standard wave equation¹¹,

\(- \partial^{2}_{0}X^{\mu} + \partial^{2}_{1}X^{\mu} = 0.\)

Surely you know how to solve that equation!

Of course, the individual components of X are not independent. They are related by the constraints.

Worse, it’s probably worth pointing out that our gauge choice doesn’t uniquely fix the section of the gauge bundle. We have a little more wiggle room to use - and worry about - later one. But we’ve seen enough for today.

Hopefully this gives you a taste for where we are going! Thanks very much for reading.

1. More precisely, invariant under diffeomorphisms of the embedding of that worldline into spacetime.

2. Three-dimensional spacetime is very interesting edge case, but nontrivial gravitational wave solutions are only present asymptotically. Without gravitational waves, there are no gravitons to quantize - and hence no interactions to renormalize.

3. Even here it’s essentially topological data, so there is no sense of kinetic energy as we’d normally encounter.

4. This is an example of a result we shall prove in another episode. Understanding its role as a function of dimension is very intriguing.

5. And we will, in gory detail, in a future episode. This seems important as all the texts from GSW to Kaku merely present this result without derivation, expecting you to know it from the study of General Relativity.

6. At least classically. The fact that it can fail quantum mechanically is the rabbit hole that makes String Theory so interesting.

7. And also to enforce the fact that we don’t want to scale the metric to zero.

8. Hermann Weyl famously studied a version of General Relativity where diffeomorphism invariance was promoted to include invariance up to local conformal transformation, which is why his name often appears with these kinds of symmetries or transformations.

9. The primary importance, of course, for selecting a globally defined section of the gauge bundle in the quantum theory.

10. Although you might recall that the mass-shell constraint was parameterized by the wordline cosmological constant, which is absent in this case. Later we shall see how the concept of a mass emerges dynamically for the string.

11. There are also boundary conditions to worry about, but again, we’ll visit these later on.